In the paper On a 1-1-correspondence between rooted trees and natural numbers by F. Goebel, a correspondence between natural numbers and rooted tree was established via prime factorization.
Let $T$ be a rooted tree, $r$ its root. The connected components of $T-r$ are denoted by $T_1,\dots,T_v$, where $v$ is the degree of $r$. The graphs $T_j$ ($j=1,\dots,v$) obviously are trees, which we transform into rooted trees by defining as the root of $T_j$ the vertex of $T_v$ which is adjacent to $r$ in $T$.
Figure 2 (below) shows all rooted trees up to n=45. I do understand all composite numbers, however I fail to understand the structure of how the rooted trees for prime numbers are structured.
Question: What are the rules to get the rooted tree for prime numbers?